# Assymetry in major scale construction

My guitar teacher recommend me to write out all the major scales. His intention was that I would see how double sharps appear in the scale. I have written them all and found this:

I wrote them following the cycle of fifths so that the sharps appear consecutively and I have put the enharmonic scales together. Assuming that they are correct, I am surprised that the maximum number of double sharps is five, while that of double flats is only one.

Does this mean, applying Occam's razor, that it is better to use flats in general and especially when there are double accidentals?

• The last four work as scales, but they won't be keys in their own rght - why would there need to be any double sharps (or double flats for that matter) in a key signature? – Tim Nov 25 '19 at 11:44
• If you move in the other direction I do not see how you won't eventually get more double flats. – ggcg Nov 25 '19 at 12:57
• I can't see a lot of point in the exercise. Double # and b are not really used in that way. If there's a # in the key sig. and that note needs raising another semitone, then that's where x comes in, rather than just being in a scale. Ask exactly why he recommended it. – Tim Nov 25 '19 at 15:50
• @ggcg I do not understand how I can construct the scales in the other direction. Could you give me a hint? – Raoul Kessels Nov 25 '19 at 17:35
• @Tim I will ask him next class, but still there are more double sharps than double flats which is what is puzzling me – Raoul Kessels Nov 25 '19 at 17:36

The reason you didn't get more double accidentals is because you didn't complete the exercise.

You made the arbitrary assumption that a the keynote of the scale can't be a double sharp or flat. For example B double flat major has two double flats.

If you write all the scales in the cycle of 5ths from C double flat through to C double sharp, you will get scales with up to 7 double flats and 7 double sharps.

(Also note that there are a few published pieces that use triple sharps and flats - though the only ones I know of are for keyboard instrments, not guitar)

• Wonder if those pieces with ###/bbb could have been written out in a 'sensible' key, which would probably eliminate the need for that. – Tim Nov 25 '19 at 13:15
• I shall try to construct those also, but I have the impression that then I will get more triple sharps than triple flats – Raoul Kessels Nov 25 '19 at 17:37
• @RaoulKessels: It's a verifiable fact that if you write out all scales on the circle of fifths from C-flat to C-sharp, you will get all keys up to 7 flats and 7 sharps. It's a verifiable fact that if you write out all scales on the circle of fifths from C-double-flat to C-double-sharp, you will get all keys up to 7 double-flats and 7-double-sharps. The only reason you have more double sharps is, as this answer said, because you went farther around the circle of fifths on the sharp side than on the flat side. – Athanasius Nov 25 '19 at 19:29
• @Athanasius Of course, this doesn't hold for modes of major; C♭ major has 7 flats, but C♭ minor has 10! Still a good point, though. – user45266 Nov 25 '19 at 19:33
• @user45266 - The question title is about major scale construction and implicitly that was what was assumed in this answer and in my comment. I don't know what minor scales have to do with OP's question. – Athanasius Nov 25 '19 at 19:36

This task wouldn't make sense if we don't consider the in melodic minor scales the upper tetrachord is borrowed from the parallel major key, as in normal major keys there are normally (in the root scale) no double sharps. But if you have secondary dominants or secondary viidim7 chords the double sharps will be used.

I don't think that Ockeghem was occupied with these problems but if you want the problem of avoiding double sharps by transcribing in the flat key (F#=Gb, A# minor=Bb minor etc) can be considered as an application of this thought.

A good musician shouldn't be bothered by this question: It makes sense to practice pieces in both enharmonic related keys.

Go in the other direction, descending fifths: C, F, Bb ...Cb, Fb, Bbb, Ebb, Abb.

You should end up see a similar number of double flats appearing.

Also, you might look into the harmonic uses of double sharps and flats. Like using `Fx` in `G#` minor to spell a `V` chord `D# Fx A#`. Or, `Bbb` to spell a minor `iv` chord in `Db` major `Gb Bbb Db`. That takes things out of the mostly theoretical and into the practical.

As mentioned in other comments, you didn't go all the way around. C to F# sharp-wise is 6 fifths, whereas C to Bb is only 2 fourths. It just so happens that the major scale contains more notes that are built from fifths rather than fourths (I.e. C-G-D-A-E-B are all built from fifths, and C-F is the only fourth), so you go through more natural notes before going into the sharps. If you go fourth-wards, you go into the flats much earlier.